Are we still on systems? Geez...
Non-dice system don't really appeal to me.
With a few exceptions they almost seem less intuitive and more cumbersome than game mechanics using dice.
It's not that I've never gone diceless but usually when I have I've also gone system-less for the most part. If I'm freewheeling it in some kind of interactive story exercise sense then I'm going to go full improv and not worry about rules. Of course, that isn't really playing a game is it.
And there in lies the great paradox; I don't want the rules to get in the way of my friends and I telling a cool story but I definitely want there to be rules and dice or else its just anarchy. It's role playing but it isn't really a game.
OK, to answer the question I would say I really like Golden Sky Stories. It works really well even though there isn't a dice system and the mechanics aid in giving it that Studio Ghibli/Hayao Miyazaki feel.
I also really like Castle Falkenstein but to be honest I liked it more for its setting and presentation. I wish it had more traditional mechanics.
I've always wanted to like a dice and playing card combo, such as you see in some indie Wild West games but they rarely work perfectly or get the feel just right.